Available in electronic format
Available in print format		 Transacrions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

On Siegel modular forms of half-integral weights and Jacobi forms

Author(s): Koichi Takase
Journal: Trans. Amer. Math. Soc. 351 (1999), 735-780.
MSC (1991): Primary 11F37, 11F27; Secondary 11F70
Retrieve article in: PDF DVI PostScript
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: We will establish a bijective correspondence between the space of the cuspidal Jacobi forms and the space of the half-integral weight Siegel cusp forms which is compatible with the action of the Hecke operators. This correspondence is based on a bijective correspondence between the irreducible unitary representations of a two-fold covering group of a symplectic group and a Jacobi group (that is, a semidirect product of a symplectic group and a Heisenberg group). The classical results due to Eichler-Zagier and Ibukiyama will be reconsidered from our representation theoretic point of view.

References:

[E-Z]
M.Eichler and D.Zagier, Theory of Jacobi Forms, Progress in Math. 55, 1985. MR 86j:11043

[God]
Godement,R., Notes on Jacquet-Langlands' theory, The Institute for Advanced Study, 1970.

[H-S]
Hecht,H. and Schmid,W., On Integrable Representations of a Semisimple Lie Group, Math.Ann. 220 (1976), 147-149. MR 53:3203

[Ibu]
Ibukiyama,T., On Jacobi forms and Siegel modular forms of half integral weights, Comment. Math. Univ. St. Pauli 41 (1992), 109-124. MR 94a:11069

[Igu]
Igusa,J.-I., Harmonic analysis and theta-functions, Acta Math. 120 (1968), 187-222. MR 43:2428

[Knp]
Knapp,A.W., Representation Theory of Semisimple Groups, Princeton Univ. Press, 1986. MR 87j:22022

[K-V]
Kashiwara,M. and Vergne,M., On the Segal-Shale-Weil Representations and Harmonic Polynomials, Inv.Math. 44 (1978), 1-47. MR 57:3311

[Kug]
Kuga,M., Fibre varieties over a symmetric space whose fibres are abelian varieties., Algebraic Groups and Discontinuous Subgroups P. S. P. M. vol. 9 (Amer. Math. Soc.) (1966), 338-346. MR 34:5990

[Mum]
Mumford,D., Families of abelian varieties, Algebraic Groups and Discontinuous Subgroups P.S.P.M. vol.9 (Amer.Math.Soc.) (1966), 347-351. MR 34:5828

[Mur]
Murase,A., $L$-functions attached to Jacobi forms of degree $n$, Part I. The basic identity, J. reine angew. Math. 401 (1989), 122-156. MR 90k:11064

[Sat1]
Satake,I., Factors of Automorphy and Fock Representations, Advances in Math. 7 (1971), 83-110. MR 50:541

[Sat2]
Satake,I., Unitary representations of semi-direct product of Lie groups of $\bar {\partial }$-cohomology, Math. Ann. 190 (1971), 177-202. MR 45:5274

[Sat3]
Satake,I., Theory of spherical functions on reductive algebraic groups over $p$-adic fields, Publ. Math., I.H.E.S. 18 (1963), 5-69. MR 33:4059

[Sat4]
Satake,I., Holomorphic imbedding of symmetric domains into Siegel space, Amer. J. Math. 87 (1965), 425-461. MR 33:4326

[Sat5]
Satake,I., Algebraic Structures of Symmetric Domains, Iwanami Shoten and Princeton Univ. Press, 1980. MR 82i:32003

[Shn]
Shintani,T., unpublished note.

[Tak1]
Takase,K., A note on automorphic forms, J.reine angew.Math. 409 (1990), 138-171. MR 91h:11040

[Tak2]
Takase,K., On unitary representation of Jacobi group, J. reine angew. Math. 430 (1992), 139-149 (Correction: J. reine angew. Math. 441 (1993), 215-216). MR 93e:11064; CMP 93:15

[Tak3]
Takase,K., On Two-fold covering group of $Sp(n,\mathbb{R})$ and automorphic factor of weight $1/2$, Comment. Math. Univ. Sancti Pauli 45 (1996), 117-145. MR 98c:11041

[Tam]
Tamagawa,T., On Selberg's trace formula, J.Fac.Sci.Univ.Tokyo 8 (1960), 363-386. MR 23:A958

[Wei]
Weil,A., Sur certains groupes d'opérateurs unitaires, Acta Math. 111 (1964), 143-211. MR 29:2324

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 11F37, 11F27, 11F70

Retrieve articles in all Journals with MSC (1991): 11F37, 11F27, 11F70

Additional Information:

Koichi Takase
Affiliation: Department of Mathematics, Miyagi University of Education, Aoba-ku, Sendai 980, Japan
Email: f26508@cctu.cc.tohoku.ac.jp

DOI: 10.1090/S0002-9947-99-02168-6
PII: S 0002-9947(99)02168-6
Received by editor(s): February 5, 1997
Copyright of article: Copyright 1999, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google